Spring 2017

As alternatives to partial differential equations (PDEs), nonlocal continuum models given in integral forms avoid the explicit use of derivatives and allow solutions to exhibit desired singular behavior. We present in this talk nonlocal models of mechanics and diffusion processes characterized by a horizon parameter which measures the range of nonlocal interactions. Considering their close connections to classical local PDE models in the limit when the horizon parameter shrinks to zero and to global fractional PDEs in the limit when the horizon parameter tends to infinity, we present numerical schemes that are robust under the changes of the horizon parameter. We also discuss the coupling of models characterized by different scales of horizon.

Geometric constraints can profoundly affect pattern selection and topological defect formation in equilibrium and non-equilibrium systems. In this talk, I will summarize recent experimental and theoretical work that aims to understand (i) how substrate curvature controls symmetry breaking and defect statistics in elastic surface crystals, and (ii) how confinement geometry affects spontaneous flows of microbial suspensions. Our results show that minimal higher-order PDE models can accurately capture the experimentally observed pattern formation transitions in these systems. We first describe phenomenological parallels between 2D elastic and colloidal crystals on spherical surfaces that suggest some universality in the underlying nucleation processes. Subsequently, we demonstrate how microbial flow patterns can be controlled by microstructure to realize bacterial spin lattices. Building on these insights, we will propose designs of active flow networks to implement logical operations in autonomous microfluidic transport devices.

Particle-based stochastic reaction-diffusion (PSRD) models have become a popular tool for modeling cellular processes in which both noise in chemical reactions and the spatial transport of molecules are important. I will briefly motivate our interest in such models by discussing the influence of volume exclusion due to organelles on the propagation of signals within cells, and by exploring a simple receptor signaling system where only spatial, stochastic models can correctly reproduce experimentally observed dynamics.

I will then introduce a general formulation of PSRD models that encompasses several of the most widely-used variants. To numerically solve a subset of these models, a hybrid finite element / finite volume discretization that has the form of a master equation for a jump process will be constructed. Exact realizations of the resulting jump process can be generated through the well-known stochastic simulation algorithm, enabling the accurate simulation of general PSRD models. Time-permitting, I will illustrate our method on several biological examples.

The statistical mechanics of energy landscapes has resolved the paradoxes of how information-bearing matter can assemble itself spontaneously. I will explain how our current understanding of protein folding landscapes not only leads to successful schemes for predicting protein structure from sequence but also has given quantitative insight into how folding and function shape molecular evolution. While protein folding is, in the main, thermodynamically controlled and not kinetically limited, longer structures in the cell can assemble in a kinetically controlled, nonequilibrium fashion. Nevertheless, I will show how energy landscape theory provides tools for extracting from low resolution experimental structural methods and kinetic information about the structure and cooperative dynamics of chromosomes.

How to extract information from exponentially growing recorded neuronal data is a great scientific challenge. It is urgent to develop methods to simplify the analysis of neuronal data. In this talk, we address what kind of dynamical states of neuronal networks allows us to have an effective description of coding schemes. For asynchronous neuronal networks, when considering the probability increment of a neuron spiking induced by other neurons, we found a probability polling (p-polling) state that captures the neuronal interactions which are affected by multiple factors, i.e., coupling structure, background input and external input. We show that this state is confirmed in some experiments in vitro and in vivo, and also confirmed through the simulation of Hodgkin-Huxley neuronal networks. We hypothesize that this p-polling state may be a general operating state of neuronal networks. For the p-polling state, we show that neuronal firing patterns can be well captured by the 2nd order maximum entropy model.

We consider the use of integral equations as nonlocal models for diffusion. An important example are fractional Laplacian equations although our theory and numerics apply to problems that cannot be modeled by fractional derivative models. We discuss theories for the analysis and numerical analysis of the models considered, relying on a nonlocal vector calculus to define weak formulations in function space settings. We also briefly discuss the nonlocal, continuum peridynamics model for solid mechanics. [Based on joint works with Marta D’Elia, Qiang Du, Richard Lehoucq, Xiaochuan Tian, and F. Xu.]

Mar. 3, Marija Vucelja, Department of Physics, University of VirginiaModelling the emergence of clones in populations (by drawing analogies with the physics of glasses) and the adaptive immune system of bacteria (called CRISPR)

I will describe about two examples where drawing analogies with physics has been pivotal to our understanding of the population genetics phenomena. First, I will talk about the emergence of clones in populations. Recombination reshuffles genetic material, while selection amplifies the fittest genotypes. If recombination is more rapid than selection, a population consists of a diverse mixture of many genotypes. In the opposite regime selection can amplify individual genotypes into large clones causing the “clonal condensation”. I will point out the similarity between clonal condensation and the freezing transition in the Random Energy Model of spin glasses. I will derive one of the key quantities of interest: the probability that two individuals are genetically identical. As my second example I will speak about the CRISPR mechanism which serves as an adaptive defense mechanism of bacteria against phages. It takes parts of genomic sequences from the ”invaders” and in this way builds up a memory of past infections. With a new encounter of an invading sequence, this memory is accessed, and in a successful outcome, the invader is neutralized. I will introduce a population dynamics model where immunity can be both acquired and lost. Two key parameters of the model are the ease of acquisition and the effectiveness in conferring immunity.

I will present three problems on biophysics, low-cost diagnostic devices, and microfluidics from my research at the Prakash Lab at Stanford. First, I will talk about a mathematical “T-swimmer” model, based on slender-body theory, that we developed to study how submillimetre-scale parasites swim in freshwater to infect humans causing schistosomiasis, a disease comparable to malaria in global socio-economic impact. Juxtaposing this model with biological experiments and a robotic realization, I will show how these parasites break time-reversal symmetry and propagate at an optimal regime for efficient swimming. Second, I will describe an ultralow-cost (20 cents), lightweight (2 g), human-powered paper centrifuge designed on the basis of a mathematical model of a nonlinear, non-conservative oscillator inspired by the ancient whirligig toy. Our centrifuge achieves speeds of 125,000 r.p.m., separates pure plasma from whole blood in less than 1.5 min and isolates malaria parasites in 15 min. Finally, I will talk about a microfluidic platform that performs universal logic operations with droplets. Through a reduced-order model and scaling laws for understanding the underlying physics, I will demonstrate droplet-based AND, OR, XOR, NOT and NAND logic gates, fanouts, a full adder, a flip-flop and a finite-state machine.

Mar. 24, Steve Wise, Department of Mathematics, University of Tennessee, KnoxvilleDiffuse Interface Models of Multi-Phase Flows with Large Density Contrast and their Efficient Approximation with Adaptive Multigrid Methods

In many fluid flow problems, two or more constituents are fully or partially immiscible, leading to multiple phases separated by interfaces. Of these, several have the additional property that at least one of the phases has a mass density that is significantly larger or smaller than the others. I will describe two problems of this type, one coming from the evaporation of a solvent in the processing of multi-phase polymer mixtures in an organic photovoltaic (OPV) application and another from simpler two-phase flows. There are many ways to build models of these phenomena. I will develop the OPV equations using the theory of mixtures, using a volume fraction approach. All of the models that I will investigate are of the diffuse interface type, wherein interfaces are approximated with a larger-than-physical interfacial thickness via an order parameter. The utility of the approach is that one does not have to explicitly track the dynamics of the interfaces. On the other hand, one does need to resolve time-evolving fields with large gradients. In the latter part of the talk, time permitting, I will describe efficient adaptive nonlinear multigrid methods for efficiently solving the model equations. I will focus on some recent advances in the adaptive algorithms for seamlessly incorporating mass conservation.

When two drops coalesce into one, the overall surface area is reduced and the released surface energy can be converted to kinetic energy, leading to self-propelled jumping motion of the merged drop. In this talk, we will first illustrate the physical mechanisms of the self-propelled motion by experimentally validated simulations. We will then discuss a few applications of the coalescence-induced motion in biology and engineering, including the launching mechanism of fungal ballistospores and the adaptive cooling of microelectronic hotspots.

The recognition of the role hemodynamic forces have in the
localization and development of disease has motivated large-scale efforts
to enable patient-specific simulations. When combined with computational
approaches that can extend the models to include physiologically accurate
hematocrit levels in large regions of the circulatory system, these
image-based models yield insight into the underlying mechanisms driving
disease progression and inform surgical planning or the design of next
generation drug delivery systems. Building a detailed, realistic model of
human blood flow, however, is a formidable mathematical and computational
challenge. The models must incorporate the motion of fluid, intricate
geometry of the blood vessels, continual pulse-driven changes in flow and
pressure, and the behavior of suspended bodies such as red blood cells. In
this talk, I will discuss the development of HARVEY, a parallel fluid
dynamics application designed to model hemodynamics in patient-specific
geometries. I will cover the methods introduced to reduce the overall
time-to-solution and enable near-linear strong scaling on up to 1,572,864
core of the IBM Blue Gene/Q supercomputer. Finally, I will present the
expansion of the scope of projects to address not only vascular diseases,
but also treatment planning and the movement of circulating tumor cells in
the bloodstream.

Fall 2016

Network analysis provides tools for addressing fundamental applications in graphs such as webpage ranking, protein-function prediction, and product categorization and recommendation. As real-world networks grow to have billions of nodes and edges, the scalability of network analysis algorithms becomes increasingly important. Whereas many standard graph analysis algorithms require exploring the entire graph, local algorithms explore only the graph region near the nodes of interest. This talk will focus on local algorithms for graph diffusions, as well as the localized behavior of global algorithms. We show how two well-studied matrix functions for graph analysis, PageRank and the matrix exponential, stay localized on networks that have a skewed degree sequence related to the power-law degree distribution common to many real-world networks. These results give the first theoretical explanation of a localization phenomenon that has long been observed in diffusions on real-world networks. We then present a sublinear local algorithm for computing a generalized class of these diffusions, and discuss uses in graph analysis tasks like community detection and node ranking.

Solvation mechanics is concerned with the, mostly electrostatic, interaction of solute molecules, such as biomolecules essential for life, with solvent molecules, water and sometimes ions. The action of solvent molecules in the thin solvation layer around solute molecules is a key determinant of behavior. We will present a range of models of varying fidelity for this process, compare to existing approaches, and discuss the usefulness of these approaches.

Sep. 23, Lev Ostrovsky, Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia, University of Colorado, Boulder, and UNC-Chapel HillAcoustic radiation force and its biomedical applications

Radiation force (RF), stress, and pressure are average forces generated by a field and acting on a body, boundary, or distributed in space. The case of
electromagnetic radiation pressure on an interface is well known and used to be a subject of intensive research in the past. The concept of Acoustic Radiation Force (ARF) was introduced by Lord Rayleigh in 1902 as “the pressure of vibrations”. More recently this concept found broad biomedical applications as a tool for manipulation of biological cells and particles, and in medical diagnostics, particularly assessing viscoelastic properties of biological tissues and in their imaging. Here we give examples of such applications, as well as of theoretical modeling of the effects of shear wave generation by an ultrasonic beam. These are used for diagnostics and manipulation of micro-particles and micro-bubbles in standing acoustic waves in plane and cylindrical configurations. Theoretical results are compared with available experimental data.

Bacteria such as E. coli swim through the fluid by utilizing their helical flagella, each of which is driven by a rotary motor. This flagellar motor is embedded in the cell body and it can turn either clockwise (CW) or counterclockwise (CCW), which will lead to either run or tumble. During a run, all of flagellar motors spin CCW while each filament forms a left-handed helix, and the hydrodynamic interaction of flagella causes the filaments to form a bundle that propels the cell forward. During a tumble, one or more flagellar motors reverse the direction of rotation and the motor reversal initiates the polymorphic transformation, which is a local change in helical pitch, radius, and the handedness.

In this work we present computational models (1) to demonstrate two mechanisms that drive polymorphic transformation by comparing to experimental data, and (2) to study flagellar bundling and unbundling. Kirchhoff rod theory is employed to describe each helical flagellum as a space curve associated with an orthonormal triad that measures the degree of bending and twisting of the rod. The elastic rod immersed in a viscous fluid applies the force and torque to the surrounding fluid and moves at the local fluid velocity whereas the triad rotates at the local angular fluid velocity.

Tracking of microscopic species is one of the most utilized experimental technologies in materials science, biophysics, tissue engineering and nanomedicine. The goal is to draw inferences (e.g., viscous and elastic moduli, mesh spacings, passage times) by statistical analysis of particle traces. This in turn allows for selection of a candidate underlying transport mechanism, or to translate mechanistic understanding to computational models with predictive power. Routinely tracked “particles” include pathogens such as viruses and bacteria, bacteriophages, passive and active microbeads, and nano-sized drug carriers. The relevant biological fluids are typically complex, such as viscoelastic and heterogeneous mucus secretions or extracelluar matrices, which often creates imaging environments with exceedingly poor signal-to-noise ratios (SNR). Worse, due to significant time and technical constraints in extracting accurate time-variant positional data from recorded movies, virtually all particle tracking experiments are performed with 2D movies that only captured the dynamics of the species of interest for very limited durations before the species swim or diffuse out of the focal plane. To overcome these shortcomings, we have developed a new approach for particle identification and tracking, based on machine learning and convolutional neural networks (CNN), a type of feed-forward artificial neural network designed to process information in a layered network of connections that mimics the organization of real neural networks in the mammalian retina and visual cortex. Our neural network tracker is capable of exceedingly accurate 3D particle tracking (<0.5% false positive rate, <10% false negative rates) even down to SNR < 1. This technical advance consequently enables high resolution 3D particle tracking, dramatically increasing the temporal duration and richness of microscopy data. The experimental-mathematical advances will dramatically reduce uncertainty in mathematical model selection and in statistical inferences of all currently tracked species, while generating novel insights into many critical physiological processes.

Oct. 14, George Biros, Institute for Computational Engineering and Sciences, The University of Texas at AustinFast algorithms for hierarchical matrices

Hierarchical matrices are matrices that have a large number of blocks that admit
a low-rank approximation. There are many ways to construct these approximations,
but the most natural ones are based on recursion. Once an approximation has been
constructed, it can be used to accelerate matrix-vector multiplications and
construct preconditioners. Operations that typically require quadratic or cubic
complexity on the size of the matrix can be reduced to linear complexity — up
to a logarithmic prefactor. Hierarchical matrices appear in computational
physics (Green’s functions) and machine learning (kernel methods). Many famous
algorithms for constructing hierarchical matrices exist: treecodes, kernel
matrices, Gram matrices, N-body methods.

In this talk, I will present some examples of applications of hierarchical
matrices in Stokesian fluid mechanics, and supervised learning. Then I will
present a review of ASKIT (approximate skeletonization kernel independent
treecode), a method that we recently developed in my group. The main feature of
ASKIT is its dependence on the local intrinsic dimension of the dataset as
opposed to the dimension of the ambient space of the dataset. We will compare
ASKIT with the Nystrom method and discuss its application to learning problems
in high-dimensions. As a highlight, we will report results for Gaussian kernel
matrices with 100 million points in 64 dimensions, and for eight million points
in 1000 dimensions.

In this talk we will show a new stabilization method for low order tetrahedral elements suitable for incompressible nonlinear implicit elastodynamics. Casting the momentum equation into a first order system, the method is based on a two-steps algorithm where we first solve the mixed P1/P1 velocity/pressure system, with the incompressibility constraint enforced through the divergence of the velocity field, and then we update the displacement field. The residual based stabilization acts only on the divergence-free equation and it constitutes a simple effective modification of the original problem leading to stable solutions. The method is derived from a variational multiscale analysis, where only the stabilizing term is maintained, and combined with several implicit time integrators. As we will show, dissipative time integrators must be used in order to get rid of unphysical high frequency pressure modes. Such modes can be appreciated only in the incompressible limit and become clear when the timestep is taken sufficiently small. The proposed algorithm can be applied to infinitesimal and finite strain deformations. In particular, we consider different classes of isotropic hyperelastic models for incompressible and nearly incompressible motion of rubber-like materials. We report several numerical examples establishing the performances and robustness of the proposed algorithm.

Immune cells exhibit stimulation-dependent traveling waves within the cortex, much faster than typical cortical actin waves. These waves reflect rhythmic assembly of both actin machinery and periphery membrane proteins such as F-BAR domain proteins. Combining theory and experiments, we develop a mechanochemical feedback model involving membrane shape changes and F-BAR domain proteins that shapes into an excitable system. We show that this excitability can only manifests itself as phase wave. That is, the spatial gradient in the timing of excitability on the cortex gives the impression of propagation. And the resulting phase speed dictates the observed fast propagation speeds. We further provide evidences that membrane shape undulations accompany such rhythms, excite further cortical activation along the wave propagation path, and potentiate curvature propagation beyond the initial cortical activation site. Therefore, membrane shape change has underappreciated roles in setting high-speed signal transduction rhythms across the entire cortex.

Nov. 11, Vakhtang Putkaradze, Departments of Mathematics and Statistics, and Chemical Engineering, University of AlbertaExact geometric approach to the discretization of fluid-structure interactions and the dynamics of tubes conveying fluid

Variational integrators for numerical simulations of Lagrangian systems have the advantage of conserving the momenta up to machine precision, independent of the time step. While the theory of variational integrators for mechanical systems is well developed, there are obstacles in direct applications of these integrators to systems involving fluid-structure interactions. In this talk, we derive a variational integrator for a particular type of fluid-structure interactions, namely, simulating the dynamics of a bendable tube conveying ideal fluid that can change its cross-section (collapsible tube). We start by deriving a fully three-dimensional, geometrically exact theory for flexible tubes conveying fluid. Our approach is based on the symmetry-reduced, exact geometric description for elastic rods, coupled with the fluid transport and subject to the volume conservation constraint for the fluid. Using these methods, we obtain the fully three dimensional equations of motion. We then proceed to the linear stability analysis and show that our theory introduces important corrections to previously derived results, both in the consistency at all wavelength and in the effects arising from the dynamical change of the cross-section. Based on this theory, we derive a variational discretization of the dynamics based on the appropriate discretization of the fluid’s back-to-labels map, coupled with a variational discretization of elastic part of the Lagrangian. Time permitting, we shall also discuss some fully nonlinear solutions and the results of experiments.

Joint work with F. Gay-Balmaz (ENS and LMD, Paris). The work was partially supported by NSERC and the University of Alberta Centennial Fund.

Over the past decade, much attention has focused on the behavior of hydrophobic particles at interfaces. This seminar will focus on the behavior of particulate rafts, which form when a monolayer of particles are placed at an air-liquid interface. The particles interact with the underlying fluid to form a quasi two-dimensional solid. Such particulate rafts can support both tension and compression, and they buckle under sufficiently large compressive loads. When a drop of surfactant is introduced into the system, fracture networks develop in the rafts. The fracture process exhibits features observed in other elastic systems, such as crack kinking, crack branching, and crack arrest. Moreover, there is a clear coupling between the raft fracture and the diffusion of the surfactant on the surface and through the “porous” liquid-particle monolayer. As such, one can draw analogies between this system and others where crack growth interacts with fluid flow or mass transport.

The seminar will present recent work in modeling the diffusion of surfactant into particle raft systems and the resulting formation of fracture networks. We will present both discrete models that track the motion of individual particles, as well as a new continuum model for poro-chemo-elasticity. Results that reproduce some of the quantitative and qualitative aspects of recent experimental studies of these systems will also be shown.

HIV remains a public health problem worldwide, with 36.7 million people currently living with HIV and 2.1 million becoming newly infected each year. Antiretroviral therapy for HIV-infected people is highly effective in preventing onward transmission at the individual level, but optimal translation of this prevention benefit to population-level reductions in HIV incidence has not been achieved. In this seminar, I will present a series of ODE models of HIV transmission in Malawi and North Carolina, each of which aims to inform public health interventions for maximizing HIV prevention in these settings.

The lattice Boltzmann method (LBM) is coupled with an intracellular-signaling-pathway model of E. coli chemotaxis to simulate biased random walk motion of chemotactic particles toward chemical stimuli in fluids while explicitly accounting for particle-fluid hydrodynamics. As part of the coupled model, the LBM allows the consideration of geometrically complex flow domains with interior obstacles and spatially- and temporally-varying chemoattractants. To better describe the biofluidic environment in the context of targeted drug delivery for cancer treatments, the model is upgraded to simulate non-Newtonian fluid flows by relating the local kinematic viscosity of the fluid to the second invariant of the rate of the shear strain tensor. The coupled algorithm can be developed further by combining the LBM and the Immersed Boundary Method to simulate the fate and transport of self-propelled deformable chemotactic particles in stagnant or flowing biofluids.

Tuesday Feb. 23, Aaron Clauset, University of Colorado Boulder, hosted by Peter MuchaInferring community structure in networks with metadata

For many networks of scientific interest we know both the connections of the network and information about the network nodes, such as the age or gender of individuals in a social network, geographic location of nodes in the Internet, or cellular function of nodes in a gene regulatory network. In this talk, I’ll show how this “metadata” can be used to improve our analysis and understanding of network structure. I’ll focus in particular on the problem of community detection in networks and present a mathematically principled approach, based on a generalization of the stochastic block model, that combines a network and its metadata to detect communities more accurately than can be done with either alone. Crucially, the method does not assume that the metadata are correlated with the communities we are trying to find. Instead the method learns whether a correlation exists and correctly uses or ignores the metadata depending on whether they contain useful information. The learned correlations are also of interest in their own right, allowing us to make predictions about the community membership of nodes whose network connections are unknown. I’ll demonstrate the model on synthetic networks with known structure, where the method performs better than any algorithm without metadata, and on real-world networks, large and small, drawn from social, biological, and technological domains.

I will discuss two fluid-mechanics problems exploited by biological systems.

First, animals that drink must transport water into the mouth using either a pressure-driven (suction) or inertia-driven (lapping) mechanism. Previous work on cats shows that these mammals lap using a fast motion of the tongue with relatively small acceleration (~1g), in which gravity is balanced with steady inertia in the liquid. Do dogs employ the same physical mechanism to lap? To answer this question, we recorded 19 dogs while lapping and conducted physical modeling of the tongue’s ejection mechanism. In contrast to cats, dogs accelerate the tongue upward quickly (~1-4g) to pinch off the liquid column. The amount of liquid extracted from the column depends on whether the dog closes the jaw before or after the pinch-off. Our recordings show that dogs close the jaw at the moment of pinch-off time, enabling them to maximize volume per lap.

In nature, several seabirds (e.g. Gannets and Boobies) dive into water at up to 24 m/s as a hunting mechanism; moreover, Gannets and Boobies have a long slender neck, which can potentially be the weakest part of the body under compression due to high-speed impact. In this work, we investigate the stability of the bird’s neck during plunge-diving by understanding the hydrodynamic forces on the cone-shaped head and the flexibility of the long neck. Firstly, we use a salvaged bird to resolve plunge-diving phases and the skull and neck anatomical features to generate a 3D-printed skull and to quantify the effect of the neck musculature to provide the necessary stability. Secondly, physical experiments of an elastic beam as a model for the neck attached to a skull-like cone revealed the limits for the stability of the neck during the bird’s dive as a function of impact velocity and geometric factors. We find that the small angle of the bird’s beak and the muscles in the neck predominantly reduce the likelihood of injury during a high-speed plunge-dive. Finally, we discuss maximum diving speeds for humans using our results to elucidate injury avoidance.

Apr. 15, Darren Narayan, RIT, hosted by Peter MuchaRefining the clustering coefficient for analysis of social and neural network data

We will show how a deeper analysis of the clustering coefficient in a network can be used to assess functional connections in the human brain. Our metric of clustering centrality considers the frequency at which an edge appears across all local subgraphs that are induced by each vertex and its neighbors. This analysis is tied to a problem from structural graph theory in which we seek the largest subgraph that is a Cartesian product of two complete bipartite graphs K_{1,m} and K_{1,1}. We investigate this property and compare it to other known edge centrality metrics. Finally, we apply the property of clustering centrality to an analysis of functional MRI data from the Rochester Center for Brain Imaging at the University of Rochester.

Fall 2015

Sep. 04, Yue Wu, University of North Carolina, hosted by Greg ForestComplex Behaviors of Simple Liquids: Nontrivial Order Parameters and Cooperativity

Liquids, even simple liquids, exhibit intriguing phenomena that defy our understanding. For instance, liquids, including monatomic liquids1, undergo glass transition when supercooled and the nature of glass transition remains unknown. Another related issue is whether liquid can undergo liquid-liquid transition (LLT), namely, a liquid of the same composition could assume different structures and coexist at given temperature and pressure. In fact, such LLT has been predicted in supercooled water. Although it is easy to understand the immiscibility of oil and water, it is thought provoking to envision two liquid phases of the same composition (such as pure water) that are immiscible. I will discuss our recent work on an observation of LLT in a metallic liquid system and molecular dynamic simulation reveals that bond-orientational order plays a critical role in such LLT instead of the conventional order parameter density2. Complex behaviors of liquids also arise under nanoconfinement including water and aqueous electrolytes3. Here again, cooperativity plays an important and could lead to bizarre behaviors with important implications for nanotechnology such as energy storage devices. These studies demonstrate the essential need of combining experimental studies with simulations in understanding the complex behaviors of liquids.

Sep. 11, Daphne Klotsa, University of North Carolina, hosted by Sorin MitranSpheres form strings and a swimmer from a spring

Rigid spherical particles in oscillating fluid flows form interesting patterns as a result of fluid mediated interactions. Here, through both experiments and simulations, we show that two spheres under horizontal vibration align themselves at right angles to the oscillation and sit with a gap between them, which scales in a non-classical way with the boundary layer thickness. A large number of spherical particles form strings perpendicular to the direction of oscillation. Investigating the details of the interactions we find that the driving force is the nonlinear hydrodynamic effect of steady streaming. We then design a simple swimmer (two-spheres-and-a-spring) that utilizes steady streaming in order to propel itself and discuss the nature of the transition at the onset of swimming as the Reynolds number gradually increases.

Asymptotic models for internal wave motion in 1+1 dimensions include nonlocal linear dispersion terms arising from the elimination of potential flow on one side of the interface via a Dirichlet-Neumann map. Such models include the intermediate long wave equation in the case of finite depth and the Benjamin-Ono equation in the case of infinite depth (of the lower fluid layer). In some situations it is physically reasonable to assume that the dispersive effects are formally small compared with nonlinear effects that eventually lead to wave breaking, and then it is interesting to study the effect that weak dispersion has as a regularizing effect on the breaking waves. This problem has been studied for many years in the context of the Korteweg-de Vries equation, with key ideas going back to Whitham, Gurevich-Pitaevskii, and Lax-Levermore, and with more modern developments such as the results of Claeys and Grava arising from the Deift-Zhou steepest descent method for Riemann-Hilbert problems. In this talk, I will describe some of our attempts to study the corresponding problem in the context of the Benjamin-Ono equation. In particular, we will present a simple and intuitive weak convergence result (joint work with Z. Xu) that is a consequence of a new analogue of the variational method of Lax and Levermore but that takes as inspiration also the moment expansion method of Wigner in random matrix theory. We will also describe new results on the direct scattering problem for rational potentials (joint work with A. Wetzel).

We show how to control an oscillator by periodically perturbing its stiffness, such that its amplitude follows an arbitrary positive smooth function. This also motivates the design of circuits that harvest energies contained in infinitesimal oscillations of ambient electromagnetic fields. To overcome a key obstacle, which is to compensate the dissipative effects due to finite resistances, we propose a theory that quantifies how small/fast periodic perturbations affect multidimensional systems. This results in the discovery of a mechanism that we call super-parametric resonance, which reduces the resistance threshold needed for energy extraction based on coupling a large number of RLC circuits.

We discuss a geometry-based statistical learning framework for performing model reduction and modeling of data sets as well as of certain classes of stochastic high-dimensional dynamical systems. We start by discussing the problem of dictionary learning for data, and introduce a new setting for the problem and a solution based on hierarchical low-rank representation of the data, together with the corresponding statistical guarantees. We then discuss how to perform other statistical learning tasks, such as regression and estimation of distributions of the data, using the learned dictionaries.We will then discuss the approximation of certain classes of stochastic dynamical systems: we assume only have access to a (large number of expensive) simulators that can return short simulations of high-dimensional stochastic system, and introduce a novel statistical learning framework for learning automatically a family of local approximations to the system, that can be (automatically) pieced together to form a fast global reduced model for the system, called ATLAS. ATLAS is guaranteed to be accurate (in the sense of producing stochastic paths whose distribution is close to that of paths generated by the original system) not only at small time scales, but also at large time scales, under suitable assumptions on the dynamics. We discuss applications to homogenization of rough diffusions in low and high dimensions, as well as relatively simple systems with separations of time scales, and deterministic chaotic systems in high-dimensions, that are well-approximated by stochastic differential equations.

Nov. 6, Kathleen Gates, University of North Carolina, hosted by Peter MuchaClustering individuals based on temporal processes

Individuals often differ in their brain processes across time, giving rise to the need for individual-level models. This heterogeneity in brain processes occurs even within predefined, arbitrary, categories (such as those based on diagnoses or gender), indicating that these groups by be further classified or perhaps are better classified along a different dimension than the one chosen by the researcher. These results suggest a need for approaches which can accommodate individual-level heterogeneity as well as classify individuals based on similarities in brain processes. In this presentation I’ll introduce a method for clustering individuals based on their dynamic processes. The approach builds from unified structural equation modeling (also referred to as structural vector autoregression), which is a statistical method for estimating effects among variables (e.g., brain regions) across time. Here, unsupervised classification of individuals occurs using community detection during the data-driven model selection procedure for arriving at individual-level models. Our approach has several advantages over existing methods for classifying individuals. In particular, it places no assumption on homogeneity of predefined subgroups within the sample and utilizes individual-level parameters that have been shown to be more reliable than some competing approaches. This flexible analytic technique will be illustrated with a simulation study and empirical functional MRI data.

We review recent advances in particle simulations, continuum theory and multiscale modeling for equilibrium and transport properties of Coulomb many-body systems in soft matter and electrochemical energy devices. The properties of dielectric and correlation effects near material interfaces are discussed under image-charge based Monte Carlo simulations. These properties are also modeled by modified Poisson-Nernst-Planck /Poisson-Boltzmann equations incorporated with a Green’s function governed by a generalized Debye-Hückel equation. Numerical methods for both particle simulations and continuum equations are present with a comparison to show attractive performance of the new model and numerical algorithms.

Dynamical systems form the basic modeling framework for an enormous variety of complex systems. Direct numerical simulation of the correspondingly complex dynamical systems is one of few means available for accurate prediction of the associated physical phenomena. However, the ever increasing need for improved accuracy requires the inclusion of ever more detail in the modeling stage, leading inevitably to ever larger-scale, ever more complex dynamical systems that must be simulated.Simulations in such large-scale settings can be overwhelming and make unmanageably large demands on computational resources; this is the main motivation for model reduction, which has as its goal the production of much simpler dynamical systems retaining the same essential features of the original systems (high fidelity emulation of input/output response and conserved quantities, preservation of passivity, etc.).I will give a brief overview of the objectives and methodology of model reduction, focussing eventually on projection methods that are both simple and capable of providing nearly optimal reduced models in some circumstances. These methods provide a framework for model reduction that allows retention of special model structure such as parametric dependence, port-Hamiltonian structure, and internal process/propagation delays.